Nearly tight approximation algorithm for (connected) Roman dominating set

A Roman dominating function of graph G is a function r : V (G) -> {0, 1, 2} satisfying that every vertex v with r(v) = 0 is adjacent to at least one vertex u with r(u) = 2. The minimum Roman dominating set problem (MinRDS) is to compute a Roman dominating function r that minimizes the weight Sigma(v is an element of V) r (v). The minimum connected Roman dominating set problem (MinCRDS) is to find a minimum weight Roman dominating function r(c) such that the subgraph of G induced by D-R = {v is an element of V vertical bar r(c)(v) = 1 or r(c) (v) = 2} is connected. In this paper, we present a greedy algorithm for MinRDS with a guaranteed performance ratio at most H(delta(max) + 1), where H(.) is the Harmonic number and delta(max) is the maximum degree of the graph. For any epsilon > 0, we show that there exists a greedy algorithm for MinCRDS with approximation ratio at most (1 + epsilon) In delta(max) + O(1). The challenge for the analysis of the MinCRDS algorithm lies in the fact that the potential function is not only non-submodular but also non-monotone.